Fuzzy Quasi-Uniform Entropy: Extending Topological Entropy to Fuzzy Quasi-Metric Spaces

Abstract

In this paper, we extend the notions of uniform and quasi-uniform entropy for uniformly continuous self-maps to the setting of fuzzy quasi-metric spaces. We introduce a new dynamical invariant, called fuzzy quasi-uniform entropy, which simultaneously accounts for asymmetry inherent in quasi-metrics and imprecision arising from fuzzy structures. Working within the framework of uniformly continuous self-maps on fuzzy quasimetric spaces, we define this entropy using Bowen-type fuzzy quasi-balls and establish its fundamental properties. Several classical results from uniform and quasi-uniform entropy theory are shown to extend naturally to this fuzzy quasi-metric context. In particular, under a suitable restriction on the fuzziness scale parameter, the fuzzy quasi-uniform entropy dominates the classical quasi-uniform entropy, thereby capturing finer distinctions in orbit complexity. We further prove that this entropy is monotone with respect to the fuzziness parameter, invariant under topological conjugacy, and converges to classical topological entropy in the metric case. Illustrative examples demonstrate the effectiveness of the theory, including situations where fuzzy quasi-uniform entropy detects nontrivial dynamical behavior even when classical quasi-uniform entropy vanishes. These results establish fuzzy quasi-uniform entropy as a robust extension of existing entropy notions for dynamical systems exhibiting asymmetry, uncertainty, or lack of metric symmetry.